Confidence Interval Estimate Mean Unknown

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Confidence Interval Estimate Mean (? Unknown)
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Confidence Interval Estimates
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Confidence Interval Mean (? Unknown)

• Assumptions
• Population standard deviation is unknown
• Population must be normally distributed

• Use Students tdistribution

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Students t Distribution
Standard Normal
Bell-Shaped Symmetric Fatter Tails
t (df 13)
t (df 5)
Z
t
0
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Degrees of Freedom (df)

• Number of observations that are free to vary
  after sample statistic has been calculated
• Example
• Sum of 3 numbers is 6X1 1 (or any
  number)X2 2 (or any number)X3 3
  (cannot vary)Sum 6

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Students t Table
Assumen 3df n - 1 2? .10?/2 .05
2.920
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Confidence Interval Mean (? Unknown)
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Estimation Example Mean (? Unknown)
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Thinking Challenge

• Youre a time study analyst in manufacturing.
  Youve recorded the following task times (min.)
  3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90
  confidence interval estimate of the population
  mean task time?

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Confidence Interval Solution

• x 3.7
• s 3.8987

• n 6, df n - 1 6 - 1 5
• t.05 2.015

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Confidence Interval Estimate of Proportion
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Confidence Interval Estimates
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Confidence Interval Proportion

• Assumptions
• Random sample selected
• Normal approximation can be used if
  

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Estimation Example Proportion

• A random sample of 400 graduates showed 32 went
  to graduate school. Set up a 95 confidence
  interval estimate for p.

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Thinking Challenge

• Youre a production manager for a newspaper. You
  want to find the defective. Of 200 newspapers,
  35 had defects. What is the 90 confidence
  interval estimate of the population proportion
  defective?

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Confidence Interval Solution
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Finding Sample Sizes
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Finding Sample Sizes for Estimating ?
SE Sampling Error
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Sample Size Example

• What sample size is needed to be 90 confident
  the mean is within ? 5? A pilot study suggested
  that the standard deviation is 45.

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Finding Sample Sizes for Estimating p
SE Sampling Error
If no estimate of p is available, use p q .5
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Sample Size Example

• What sample size is needed to estimate p with 90
  confidence and a width of .03?

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Thinking Challenge

• You work in Human Resources at Merrill Lynch.
  You plan to survey employees to find their
  average medical expenses. You want to be 95
  confident that the sample mean is within 50.
  A pilot study showed that ? was about 400.
  What sample size do you use?

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Sample Size Solution
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Statistics for Business and Economics

• Chapter 6
• Inferences Based on a Single Sample Tests of
  Hypothesis

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Learning Objectives

• Distinguish Types of Hypotheses
• Describe Hypothesis Testing Process
• Explain p-Value Concept
• Solve Hypothesis Testing Problems Based on a
  Single Sample
• Explain Power of a Test

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Statistical Methods
StatisticalMethods
InferentialStatistics
DescriptiveStatistics
HypothesisTesting
Estimation
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Hypothesis Testing Concepts
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Hypothesis Testing
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Whats a Hypothesis?

• A belief about a population parameter
• Parameter is population mean, proportion,
  variance
• Must be stated before analysis

I believe the mean GPA of this class is 3.5!
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Null Hypothesis

• What is tested
• Has serious outcome if incorrect decision made
• Always has equality sign ?, ?, or ??
• Designated H0 (pronounced H-oh)
• Specified as H0 ? ? some numeric value
• Specified with sign even if ? or ??
• Example, H0 ? ? 3

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Alternative Hypothesis

• Opposite of null hypothesis
• Always has inequality sign ?,??, or ?
• Designated Ha
• Specified Ha ? ?,??, or ? some value
• Example, Ha ? lt 3

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Identifying HypothesesSteps

• Example problem Test that the population mean is
  not 3
• Steps
• State the question statistically (? ? 3)
• State the opposite statistically (? 3)
• Must be mutually exclusive exhaustive
• Select the alternative hypothesis (? ? 3)
• Has the ?, lt, or gt sign
• State the null hypothesis (? 3)

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What Are the Hypotheses?

Is the population average amount of TV viewing 12
hours?

• State the question statistically ? 12
• State the opposite statistically ? ? 12
• Select the alternative hypothesis Ha ? ? 12
• State the null hypothesis H0 ? 12

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What Are the Hypotheses?

Is the population average amount of TV viewing
different from 12 hours?

• State the question statistically ? ? 12
• State the opposite statistically ? 12
• Select the alternative hypothesis Ha ? ? 12
• State the null hypothesis H0 ? 12

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What Are the Hypotheses?

Is the average cost per hat less than or equal to
20?

• State the question statistically ? ? 20
• State the opposite statistically ? ? 20
• Select the alternative hypothesis Ha ? ? 20
• State the null hypothesis H0 ? ? 20

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What Are the Hypotheses?

Is the average amount spent in the bookstore
greater than 25?

• State the question statistically ? ? 25
• State the opposite statistically ? ? 25
• Select the alternative hypothesis Ha ? ? 25
• State the null hypothesis H0 ? ? 25

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Basic Idea
... therefore, we reject the hypothesis that ?
50.
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Level of Significance

• Probability
• Defines unlikely values of sample statistic if
  null hypothesis is true
• Called rejection region of sampling
  distribution
• Designated ??(alpha)
• Typical values are .01, .05, .10
• Selected by researcher at start